In this paper, we call AI-StoSQP a framework for two reasons. First, AI-StoSQP is trimmed from a complete StoSQP scheme. It preserves all the features that are essential for local asymptotic analysis (i.e. \(t\rightarrow \infty \) ). However, for a complete StoSQP scheme, one may insert another step between Step 2 and Step 3 to select suitable penalty parameter for a certain merit function. The selected parameter may kick in \(\beta _t\) , \(\eta _t\)  as a multiplier. See [1]} for the usage of the \(\ell _1\)  merit function, \(f({x}) + \mu \Vert c({x})\Vert _1\) . The step of penalty parameter selection is important for global analysis, while is negligible for local analysis. This is because the penalty parameter is both upper and lower bounded by deterministic thresholds, and always stabilizes for large \(t\)  (under suitable conditions). See, for example, [1]} and [3]}. Thus, the local behavior of \(({x}_t, {\lambda }_t)\)  is fully characterized by the direction \(({\bar{\Delta }}{x}_t, {\bar{\Delta }}{\lambda }_t)\)  and the controlled sequences \(\lbrace \beta _t, \eta _t\rbrace \) , which are preserved by AI-StoSQP. (Note that the deterministic SQP has the same trim, where a unit stepsize is employed and the local (quadratic/superlinear) behavior is established by purely investigating the nature of (modified) Newton system). Second, AI-StoSQP does not suggest particular designs in Step 2 and Step 3, thus providing much flexibility. One can adopt different sketching matrices in Step 2, driven by the structure of the objective; and more importantly, can adopt different procedures to select \({\bar{\alpha }}_t\)  in Step 3. For example, [1]}, [5]}, [6]}, [7]} utilized either \(\ell _1\)  or \(\ell _2\)  merit functions, and projected a random quantity into the interval \([\beta _t, \eta _t]\)  to obtain \({\bar{\alpha }}_t\) . One can design a similar procedure for augmented Lagrangian merit function, as used in [3]}, [9]}. However, such a design would also fit in the presented framework. Thus, our analysis on AI-StoSQP is broadly applicable.
